maths.free › Numerical Methods › Root finding: bisection and Newton's method

Root finding: bisection and Newton's method

Finding where f(x) = 0 to any accuracy, and how fast each method gets there.

Bisection halves an interval that brackets a root: slow but unfailing. Newton follows the tangent line to the axis, quadratic convergence when it works. Picture it: each tangent line landing closer to the root; the points x₀, x₁, x₂ are marked on the graph. Think it: Newton is a fixed-point iteration of x − f/f′, and its speed comes from that map having zero derivative at the root.

Shembulli i punuar · newton's method on x^2 - 2 from 1

Numerical-Methods · Root-Finding

Newton's method on x^2 - 2

x^{2} - 2

Hap pas hapi

  1. f(x) = x^{2} - 2,\quad x_0 = 1

    Newton's method: x_{n+1} = x_n − f(x_n)/f′(x_n). Follow the tangent line down to the axis, repeat.

  2. f'(x) = 2 x

    Differentiate once; the same derivative serves every iteration.

  3. x_{1} = 1.0 - \frac{-1.0}{2.0} = 1.5

    Iteration 1.

  4. x_{2} = 1.5 - \frac{0.25}{3.0} = 1.41666667

    Iteration 2.

  5. x_{3} = 1.41666667 - \frac{0.00694444}{2.83333349} = 1.41421569

    Iteration 3.

  6. x_{4} = 1.41421569 - \frac{6.01 \cdot 10^{-6}}{2.82843161} = 1.41421356

    Iteration 4.

  7. x_{5} = 1.41421356 - \frac{0.0}{2.82842731} = 1.41421356

    Iteration 5.

  8. x_{6} = 1.41421356 - \frac{0.0}{2.82842731} = 1.41421356

    Iteration 6.

  9. \text{converged}

    The iterates have stopped changing to 12 decimal places.

  10. x \approx 1.41421356

    Quadratic convergence: the number of correct digits roughly doubles each step.

Zbulo përgjigjen
x \approx 1.41421356

Tani ti. Zgjidh një problem, shkruaj apo vizato tënd. Çdo hap, një imazh, përgjigja e fshehur deri sa të pyesësh.

Mbaje punën tënde

Një llogari e lirë shtohet shënime në çdo mësim, një regjistrim të asaj që ju keni përfunduar, problemet tuaja të zgjidhura në një vend, dhe një mësues që ju mund të pyesni rreth kësaj faqeje. Matematika vetë është e hapur për të gjithë, të regjistruar apo jo.

Regjistrohu Hyr

Simbolet e përdorura këtu

Prek çdo simbol për përkufizimin e plotë, një fotografi dhe se çfarë do të thotë çdo shkronjë në të.

Si: Root finding: bisection and Newton's method

  1. Bracket the root: find a and b with f(a) and f(b) of opposite sign.
  2. For bisection, halve the interval keeping the sign change; each step gains one bit.
  3. For Newton, start inside the bracket and iterate x − f(x)/f′(x).
  4. Stop when successive iterates agree to the precision you need, and confirm f is tiny there.

Pyetja që bëjnë njerëzit

Why not just solve exactly?

Most equations have no closed-form solution at all, and many that do are unusable in practice. A numerical method delivers as many correct digits as you need, and a good one tells you how many that is.

Why can Newton's method fail?

If it starts where the tangent is nearly flat it shoots far away; near a repeated root it slows to a crawl; and with several roots it may land on the wrong one. A bracketing method like bisection is slower but cannot fail.

Më shumë në Numerical Methods

Numerical integration: trapezoid and SimpsonInterpolation and Taylor approximationFloating point and error